How high school boredom led to Goskie’s national mathematical recognition
A year ago, mathematics major Max Goskie sat among his peers at Pizza and Problem Solving, a Belmont club where math enthusiasts meet biweekly to tackle equations and enjoy a slice. One night, Goskie casually mentioned a theoretical math problem that occurred to him while daydreaming during a standardized test in high school.
Goskie had not given the problem much thought since, but as he presented it to the group, he was surprised to find his fellow club members were equally intrigued — and stumped.
The Bubble Path Problem
The problem went unsolved that evening, leading Goskie to team up with his academic advisor Adam Cartisano, assistant professor of mathematics, to research the formidable equation and establish a formula to solve it throughout his sophomore year.
Goskie dubbed his theory “The Bubble Path Problem,” which posed the question ‘What is the probability of creating a clear path from the top of a bubble test sheet to the bottom without hitting a filled bubble on a test with any number of questions?’
“Think of a bubble sheet or Scantron sheet you fill out during a standardized test,” Goskie explained. “You have a bunch of rows with columns to answer A, B, C or D, and you fill in with one answer per row. Now imagine you are trying to make a right-angled path from the top to the bottom using the unfilled cells with the filled-in bubbles acting as obstacles. Sometimes you can make the path, but sometimes you get a test where four answers in a row are A, B, C and D and they form a chain that blocks the path to the bottom.”
Goskie and Cartisano met regularly throughout the Fall 2023 semester to conduct research and apply Markov chains, line algebra and other operations to develop a formula that solved the problem. The two wanted their formula to apply to any four-choice bubble test, regardless of how many questions are asked.
Their final formula, in LaTeX form, read:
$P_4(n)=-0.143(0.5)^n-0.041(-0.118)^n+0.115(0.134)^n+1.068(0.984)^n$
Presenting His Findings
Goskie presented his research at Belmont’s Science University Research Symposium that same semester, and at Cartisano’s suggestion, entered the 2024 MAA MathFest Undergraduate Poster Session in Indianapolis in August 2024.
At MathFest, Goskie delivered a four-minute presentation alongside his poster to those in attendance. His work was showcased with approximately 70 other posters, and after the judges reviewed all submissions, his stood out among the rest. His research earned an “Outstanding Poster” title, one of the most prestigious awards given at the conference.
Belmont's Supportive Role in Goskie's Endeavors
Goskie attributes his success at MathFest to the communication skills he developed at Belmont, which played a crucial role in helping him earn this prestigious recognition.
“I’ve always struggled with putting my thoughts into words, and Belmont has helped me with that a lot,” he said. “Everything I’ve learned has come from people with good communication skills, and unless math becomes an every-man-for-himself subject, we will all need to learn from someone else. That’s how we move forward.”
Goskie’s experience working alongside his professor and academic advisor showcases the attention Belmont students receive from faculty and how tight knit the College of Sciences & Mathematics (CSM) community is.
“I’m able to see the math and computer science people in the CSM lounge, and with it being smaller, I know most of them — it’s great to have that kind of community, and it extends to Pizza and Problem Solving,” he said. “I also love the math classes and how much access we have at Belmont. I have been able to take some 4000 level courses despite not being in my fourth year which I would not get at other universities.”
Goskie's path from casual observation to national recognition demonstrates how curiosity, perseverance and supportive academic environments can transform simple ideas into significant achievements. His story not only highlights his individual talent but also showcases Belmont’s commitment to fostering student growth through collaborative learning and personalized mentorship.
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